Mister Exam

Lang:

Sum of series/ cos(npi)/5^n

Sum of series cos(npi)/5^n

You have entered [src]

  oo           
____           
\   `          
 \    cos(n*pi)
  \   ---------
  /        n   
 /        5    
/___,          
n = 1

\sum_{n=1}^{\infty} \frac{\cos{\left(\pi n \right)}}{5^{n}}

Sum(cos(n*pi)/5^n, (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{\cos{\left(\pi n \right)}}{5^{n}}

It is a series of species

a_{n} \left(c x - x_{0}\right)^{d n}

- power series.
The radius of convergence of a power series can be calculated by the formula:

R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}

In this case

a_{n} = \cos{\left(\pi n \right)}

and

x_{0} = -5

d = -1

c = 0

then

\frac{1}{R} = \tilde{\infty} \left(-5 + \lim_{n \to \infty} \left|{\frac{\cos{\left(\pi n \right)}}{\cos{\left(\pi \left(n + 1\right) \right)}}}\right|\right)

\frac{1}{R} = \tilde{\infty} \left(-5 + \lim_{n \to \infty} \left|{\frac{\cos{\left(\pi n \right)}}{\cos{\left(\pi \left(n + 1\right) \right)}}}\right|\right)

R = 0 \left(-5 + \lim_{n \to \infty} \left|{\frac{\cos{\left(\pi n \right)}}{\cos{\left(\pi \left(n + 1\right) \right)}}}\right|\right)^{-1}

The rate of convergence of the power series

The answer [src]

  oo               
 ___               
 \  `              
  \    -n          
  /   5  *cos(pi*n)
 /__,              
n = 1

\sum_{n=1}^{\infty} 5^{- n} \cos{\left(\pi n \right)}

Sum(5^(-n)*cos(pi*n), (n, 1, oo))

Numerical answer [src]

-0.166666666666666666666666666667

-0.166666666666666666666666666667

The graph